A fundamental topic in joint source channel coding is the design and analysis of optimal noisy channel quantization, which generally includes two parts: a vector quantizer (VQ) and an index assignment mapping. In this context the goal of optimization is to minimize the average end-to-end distortion (EED) of the system by properly designing the VQ and index assignment mapping. Given the complete knowledge of a channel, i.e., all transitional probabilities from channel input symbols to channel output symbols, both the VQ and index assignment mapping will impact on the EED.
Previously, the design and analysis of optimal noisy channel quantization were treated separately to a large extent. For example, given the complete knowledge of a channel and a fixed index assignment mapping, two necessary conditions were derived for a VQ to minimize the EED: see H. Kumazawa, M. Kasahara, and T. Namekawa, “A construction of vector quantizers for noisy channels”, Electronics and Engineering in Japan, Vol. 67-B, No. 4, pp. 39-47, 1984. These two conditions, in general, depend on all transitional probabilities from channel input symbols to channel output symbols and the fixed index assignment mapping, making it difficult to analyze the performance of the corresponding VQ and tandem system. Indeed, the two conditions reveal no new structural information about the optimal noisy channel VQ itself beyond the conventional centroid condition and nearest neighbor condition for a noiseless channel VQ: see S. Lloyd, “Least squares quantization in PCM”, IEEE Trans. on Information Theory, Vol. IT-28, No. 2, pp. 129-137, March 1982; J. Max, “Quantizing for minimum distortion,” IRE Transactions on Information Theory, Vol. IT-6, pp. 7-12, March 1960. The performance of VQs designed by an iterative algorithm based on these two conditions (which may be referred to as the KKN algorithm or the Lloyd-Max algorithm) is very sensitive to the variation of channel conditions.
On the other hand, instead of working with a fixed index assignment mapping, Zeger and Manzella investigated the case of random index assignment, where the mapping from the quantization codebook with size N to N channel symbols is chosen from the set of all N! permutations randomly and uniformly, and derived a few high rate asymptotic performance results on noisy channel quantization: K. Zeger, and V. Manzella, “Asymptotic bounds on optimal noisy channel quantization via random coding”, IEEE Trans. on Information Theory, Vol. 40, No. 6, pp. 1926-1938, November 1994. However, the analysis was carried out based on the assumption that the VQ itself is optimized without considering the channel noise, although VQs jointly designed with the channel conditions have a performance gain over VQs optimized without reference to the channel conditions. As a result, the analysis by Zeger and Manzella did not shed light on how to design optimal noisy channel VQs jointly with the channel conditions. Other related works in joint source channel coding have studied noisy channel quantization mainly from the index assignment point of view.
It would be advantageous to provide new methods for optimizing vector quantizers for noisy channels and the quantizers resulting therefrom.
Similar reference numerals may have been used in different figures to denote similar components.